Managing risk exposures using the risk budgeting approach
Bruder, Benjamin and Roncalli, Thierry
Lyxor Asset Management, Université d’Evry val d’Essonne
January 2012
Online at https://mpra.ub.uni-muenchen.de/37749/
MPRA Paper No. 37749, posted 30 Mar 2012 12:33 UTC
the Risk Budgeting Approach ∗
Benjamin Bruder Research & Development Lyxor Asset Management, Paris
benjamin.bruder@lyxor.com
Thierry Roncalli Research & Development Lyxor Asset Management, Paris
thierry.roncalli@lyxor.com First version: January 2012 – This version: March 2012
Abstract
The ongoing economic crisis has profoundly changed the industry of asset manage- ment by putting risk management at the heart of most investment processes. This new risk-based investment style does not rely on return forecasts and is therefore assumed to be more robust. 2011 was marked by several great successes in transforming the practice of asset management with several large institutional investors moving their portfolios to minimum variance, ERC or risk parity strategies. These portfolio con- structions are special cases of a more general class of allocation models, known as the risk budgeting approach. In a risk budgeting portfolio, the risk contribution from each component is equal to the budget of risk defined by the portfolio manager. Unfor- tunately, even if risk budgeting techniques are widely used by market practitioners, there are few results about the behavior of such portfolios in the academic literature.
In this paper, we derive the theoretical properties of the risk budgeting portfolio and show that its volatility is located between those of the minimum variance and weight budgeting portfolios. We also discuss the existence, uniqueness and optimality of such a portfolio. In the second part of the paper, we propose several applications of risk budgeting techniques for risk-based allocation, like risk parity funds and strategic asset allocation, and equity and bond alternative indexations.
Keywords: risk budgeting, risk management, risk-based allocation, equal risk contribution, diversification, concentration, risk parity, alternative indexation, strategic asset allocation.
JEL classification: G11, C58, C60.
1 Introduction
The investment industry of developed countries has changed with the subprime crisis of 2008 and the ongoing crisis of sovereign debt. For example, the number of mutual funds in Europe has decreased by 5% since 2008 after a growth of 25% between 2004 and 2008 (Investment Company Institute, 2011). Retail investors’ risk aversion is ever increasing and they have turned to asset managers who had achieved to manage the financial crisis in a robust way. In the mean time, the business of hedge funds has evolved completely. It is now driven by more transparent investment vehicles, like newcits funds and managed accounts,
∗We thank seminar participants at ESSEC Business School (WG Risk) and Guillaume Weisang for their helpful comments.
and is dominated by large fund managers. Investment management for institutional clients faces the same issues:
“With strong capacities in transparency and risk management becoming just as important as overall performance, many institutional investors – insurers, pension funds, corporations, governments and other entities – have put asset managers under scrutiny” (Boston Consulting Group, 2011).
The pressure for more transparency and robustness has therefore modified the relationships between investors and portfolio managers. The time is over when a fund manager could promise the moon. Today, the job of a fund manager is first of all to manage risk.
With the lower risk tolerance of investors, a new risk-based investment style has emerged since the subprime crisis. Although the importance of risk considerations in asset allocation is widely admitted, the idea is often simplified to volatility minimization as described in classical portfolio theory. Mean-variance optimization, however, generally leads to portfolios concentrated in terms of weights. Slight differences in inputs can lead to dramatic changes in allocations and create portfolios heavily invested on very few assets. There is also confusion between optimizing the volatility and optimizing the risk diversification that could be naively described by the general “don’t put all your eggs in one basket” concept. It is not the first time that mean-variance portfolio optimization encounters criticisms from investors. In the nineties, academics have proposed to improve the Markowitz model by correcting some drawbacks of this model, such as portfolio resampling (Michaud, 1989) or robust allocation (Tütüncü and Koenig [2004]). But they have continued to keep the same framework, that is minimizing the volatility of the portfolio and targeting an expected return. What is new today is that some investors don’t use optimization methods and may prefer some heuristic solutions based on risk concentration.
This new risk-based investment style puts diversification at the heart of the investment process and corresponding strategies share the property that they don’t use any performance forecasts as inputs of the model. In a risk budgeting approach, the investor only chooses the risk repartition between assets of the portfolio, without any consideration of returns. In the equal risk contribution (ERC) portfolio, the risk contribution from each portfolio asset is made equal. This portfolio has been extensively studied by Maillard et al. (2010) who had derived several interesting properties. In particular, they have shown that this portfolio is located between minimum variance and equally-weighted portfolios. In some cases, the investor doesn’t want to manage the risk exposures in a uniform way. Risk budgets are then not equal. For example, a long-term institutional investor could invest a small part of his strategic portfolio in alternative investments like commodities1. Bruderet al. (2011) present also a method to manage sovereign bond portfolios when the risk budget for each country is proportional to the GDP of the country. Unfortunately, even if risk budgeting techniques are widely used by market practitioners, they are few results about the behavior of such portfolios in the academic literature. The goal of this paper is to fill the gap between practice and theory and to give some insights about the behavior of such techniques.
The paper is organized as follows. In section 2, we derive the theoretical properties of the risk budgeting portfolio. We show that its volatility is located between those of minimum variance and weight budgeting portfolios. We also discuss the existence and uniqueness of such a portfolio. By studying the optimality property, we obtain a simple relationship
1The risk budget of these alternative investments could then represent5%or10%of the total risk of the portfolio.
between risk contributions and performance contributions. In section 3, we propose several applications of risk budgeting techniques for risk-based allocation, like risk parity funds and strategic asset allocation, and equity and bond alternative indexations. Section 4 draws some conclusions.
2 Analysis of the risk budgeting approach
2.1 Definition
Let us consider a portfolio of n assets. We define xi as the exposure (or weight) of the ith asset and R(x1, . . . , xn) as a risk measure for the portfolio x = (x1, . . . , xn). If the risk measure is coherent and convex (Artzner et al., 1999), it verifies the following Euler decomposition:
R(x1, . . . , xn) =
n
X
i=1
xi· ∂R(x1, . . . , xn)
∂ xi
The risk measure is then the sum of the product of the exposure by its marginal risk. In this case, it is natural to define the risk contribution of theith asset as follows:
RCi(x1, . . . , xn) =xi·∂R(x1, . . . , xn)
∂ xi
We consider a set of given risk budgets{b1, . . . , bn}. Here,bi is an amount of risk measured in dollars. The risk budgeting portfolio is then defined by the following constraints:
RC1(x1, . . . , xn) =b1
...
RCi(x1, . . . , xn) =bi
...
RCn(x1, . . . , xn) =bn
(1)
It is also the portfolio such that the risk contributions match the risk budgets. Contrary to the weight budgeting portfolio2, we have to solve the system (1) of nonlinear equations to define the risk budgeting portfolio.
In this paper, we focus on the volatility risk measure3. We obtain:
R(x) = σ(x)
= √ x⊤Σx
2We notexwbthe weight budgeting portfolio defined by:
xi=bi
3We notice however that most of the results could be generalized to other risk measures (see paragraph 2.4 page 13).
It comes that the marginal risk and the risk contribution of theith asset are respectively4:
∂R(x)
∂ xi
= (Σx)i
√x⊤Σx RCi(x1, . . . , xn) = xi· (Σx)i
√x⊤Σx We check that the volatility verifies the Euler decomposition:
n
X
i=1
RCi(x1, . . . , xn) =
n
X
i=1
xi· (Σx)i
√x⊤Σx
= x⊤ Σx
√x⊤Σx
= √ x⊤Σx
= σ(x)
Remark 1 We notice that if the asset returns are Gaussian, the value-at-risk of the portfolio is:
VaR (x;α) = Φ−1(α)√ x⊤Σx whereas the expression of the expected shortfall risk measure is:
ES (x;α) = 1 (1−α)
rx⊤Σx
2π e−12(Φ−1(α))2
Managing risk exposures with value-at-risk or expected shortfall risk measures is then equiv- alent to manage risk exposures with the volatility in a Gaussian world5.
2.2 The right specification of the RB portfolios
The system (1) is too large to define a portfolio which may be interesting to analyse in an asset management point of view. First, it is difficult to think in terms of exposures and nominal risk budgets. That’s why we prefer to define the portfolio in terms of weights and the risk budgets in relative value6. Second, risk budgeting techniques are used to build diversified portfolios, and specifying that some assets have a negative risk contribution implies that the risk is highly concentrated in the other assets of the portfolio. We also prefer to obtain a long-only portfolio meaning that all the weights are positive. That’s why we specify the following mathematical system to define a proper RB portfolio:
xi·(Σx)i =bi· x⊤Σx bi≥0
xi≥0 Pn
i=1bi= 1 Pn
i=1xi= 1
(2)
4Because the vectorial derivative is:
∂R(x)
∂ x = Σx
√x⊤Σx
5Under the assumption that returns are Gaussian, this result is verified for all risk measures that will not depend on the mean of the distribution.
6It implies that two constraintsPn
i=1xi= 1andPn
i=1bi= 1have to been added to the specification of the problem.
Another difficulty may appear when one specifies that some risk budgets are equal to zero. LetΣbe the covariance matrix specified as follows:
(Σ)i,j =ρi,jσiσj
where σi > 0 is the volatility of the asset i and ρi,j is the cross-correlation7 between the assetsiandj. It comes that:
∂ σ(x)
∂ xi = xiσ2i +σiP
j6=ixjρi,jσj
σ(x)
Suppose that the risk budgetbk is equal to zero. It means that:
xk
xkσ2k+σk
X
j6=k
xjρk,jσj
= 0
We obtain two solutions. The first one isx′k= 0whereas the second one verifies:
x′′k =− P
j6=kxjρk,jσj
σk
Ifρk,j ≥0for allj, we haveP
j6=kxjρk,jσj≥0 becausexj≥0 andσj >0. It implies that x′′k ≤0 meaning that x′k = 0 is the unique positive solution. Finally the only way to have x′′k >0is to have some negative correlationsρk,j. In this case, it implies that:
X
j6=k
xjρk,jσj <0
If we consider a universe of three assets, this constraint is verified fork= 3and a covariance matrix such that ρ1,3 < 0 and ρ2,3 <0. For example, if σ1 = 20%, σ2 = 10%, σ3 = 5%, ρ1,2= 50%,ρ1,3=−25%andρ2,3=−25%, the two solutions are(33.33%,66.67%,0%)and (20%,20%,40%) if the risk budgets are(50%,50%,0%).
In practice, this second solution may not satisfy the investor. When he sets one risk budget to zero, he expects that he will not have the corresponding asset in his portfolio.
That’s why it is important to impose the strict constraintbi >0. To summarize, the RB portfolio is the solution of the following mathematical problem:
x⋆=
x∈[0,1]n:Pn
i=1xi = 1, xi·(Σx)i=bi· x⊤Σx (3) whereb∈]0,1]n andPn
i=1bi= 1.
Remark 2 If we would like to impose that some risk budgets are equal to zero, we first have to reduce the universe of assets by excluding the assets corresponding to these zero risk contributions. Then, we solve the problem (3).
2.3 Some properties of the RB portfolio
In this section, we first analyse the RB portfolio in the two-asset case and the general case.
Then, we show that the RB portfolio always exists and is unique. We also discuss the optimality of such a portfolio and illustrates what the solution becomes when some risk budgets are set to zero.
7We have of courseρi,i= 1.
2.3.1 The two-asset case (n= 2)
We begin by analyzing the RB portfolio in the bivariate case. Let ρ be the correlation, x= (w,1−w)be the vector of weights and(b,1−b)be the vector of risk budgets. The risk contributions are:
RC1
RC2
= 1 σ(x)
w2σ12+w(1−w)ρσ1σ2
(1−w)2σ22+w(1−w)ρσ1σ2
The unique solution satisfying 0≤w≤1 is:
w⋆=(b−1/2)ρσ1σ2−bσ22+σ1σ2
q
(b−1/2)2ρ2+b(1−b) (1−b)σ21−bσ22+ 2 (b−1/2)ρσ1σ2
We notice that the solution depends on the correlationρ contrary to the ERC portfolio8 (Maillard et al., 2010). It is indeed a complex function of the volatilities σ1 and σ2, the correlationρand the risk budgetb.
In order to have some intuitions about the behavior of the solution, we consider some special cases. For example, ifρ= 0, we obtain:
w⋆= σ2√
b σ1√
1−b+σ2
√b
The weight of assetiis then proportional to the square root of its risk budget and inversely proportional to its volatility. Ifρ= 1, the solution reduces to:
w⋆= σ2b σ1(1−b) +σ2b
In this case, the weight of asseti is directly proportional to its risk budget. Ifρ=−1, we notice also that the solution does not depend on the risk budgets:
w⋆= σ2
σ1+σ2
because the volatility of the portfolio is equal to zero. We may also show that the weight of the asseti is an increasing function of its risk budget and a decreasing function of its volatility. In Table 1, we report some values taken byw⋆ with respect to the parametersρ andb, whereas the behavior ofw⋆is illustrated in Figure 1.
2.3.2 The general case (n >2)
In more general cases, where n > 2, the number of parameters increases quickly, with n individual volatilities and n(n−1)/2 bivariate correlations. Because the two-asset case leads to a complex solution, finding an explicit solution for the general case is certainly impossible. But we can find some results that help us to understand the behavior of the RB portfolio.
Let us begin by assuming that the correlation matrix of assets is constant:
ρi,j=ρ
8This last one is obtained ifb=1/2:
w⋆= σ2
σ1+σ2
Table 1: Weights w⋆with respect to some values of bandρ
σ2=σ1 σ2= 3×σ1
b 20% 50% 70% 90% 20% 50% 70% 90%
−50% 41.9% 50.0% 55.2% 61.6% 68.4% 75.0% 78.7% 82.8%
0% 33.3% 50.0% 60.4% 75.0% 60.0% 75.0% 82.1% 90.0%
ρ 25% 29.3% 50.0% 63.0% 80.6% 55.5% 75.0% 83.6% 92.6%
50% 25.7% 50.0% 65.5% 84.9% 51.0% 75.0% 85.1% 94.4%
75% 22.6% 50.0% 67.8% 87.9% 46.7% 75.0% 86.3% 95.6%
90% 21.0% 50.0% 69.1% 89.2% 44.4% 75.0% 87.1% 96.1%
Figure 1: Evolution of the weightw⋆(in %) with respect tob andρ
If we have no correlation, that is ifρ= 0, the risk contribution of the assetibecomesRCi= x2iσi2/σ(x). The RB portfolio being defined byRCi =biσ(x)for alli, some simple algebra shows that this is here equivalent top
bjxiσi =√
bixjσj. Coupled with the (normalizing) budget constraintPn
i=1xi = 1, we deduce that:
xi=
√biσ−1i Pn
j=1
pbjσ−1j (4) The weight allocated to the componenti is thus proportional to the square root of its risk budget and the inverse of its volatility. The higher (lower) the volatility of a component, the lower (higher) its weight in the RB portfolio. In the case of perfect correlation, that is ρ= 1, the risk contribution of the assetibecomesRCi =xiσi
Pn j=1xjσj
/σ(x). It comes
thatbjxiσi=bixjσj. We then deduce that:
xi = biσ−1i Pn
j=1bjσj−1 (5)
The opposite of perfect correlation corresponds to the lower bound of the constant correlation matrix, which is reached forρ=−1/(n−1). In this case, the volatility of the portfolio is equal to zero and the solution is the ERC portfolio9:
xi= σ−1i Pn
j=1σ−1j (6)
Unfortunately, there is no analytical solution in the general case when the constant corre- lation is different from0, 1 and−1/(n−1). Nevertheless, we could find an implicit form.
We remind that the RB portfolio verifies the following equation in the case of the constant correlation matrix:
xiσi
(1−ρ)xiσi+ρ
n
X
j=1
xjσj
=biσ2(x)
If we note Xi = xiσi and Bi = biσ2(x), the previous equation becomes (1−ρ)Xi2+ ρXi
Pn j=1Xj
=Bi. It implies that the general form of the solution is:
xi= fi(ρ, b)σ−1i Pn
j=1fj(ρ, b)σj−1
The function fi(ρ, b) depends on the constant correlation ρ and the vector b of risk bud- gets. In particular, it verifies fi
−(n−1)−1, b
= 1, fi(0, b) = √
bi, fi(1, b) = bi and fi ρ, n−11
= 1.
To illustrate the case of the constant correlation matrix, we have reported some simu- lations in Figure 2. We consider a universe ofn assets with same volatilities. We note b1
the risk budget of the first asset. The risk budget of the other assets is uniform and equal to(1−b1)/(n−1). We notice the effect of the constant correlationρon the weightx1, in particular when the numbernof assets is large. When the number of assets is small (less than 10), the correlationρhas an impact only when it is small (less than 10%). In other cases, the formula (5) is a good approximation of the solution10.
In other cases, it is not possible to find explicit solutions of the RB portfolio. But we can find a financial interpretation of the RB portfolio. We remind that the covariance between the returns of assets and the portfoliox is equal to Σx. The beta βi of the asset i with respect to the portfolio x is then defined as the ratio between the covariance term (Σx)i and the variance x⊤Σx of the portfolio. βi indicates the sensitivity of the asset i to the
9The proof is provided in Appendix A.1.
10Letxi(ρ)be the weight of the asset iin the RB portfolio when the constant correlation is equal toρ.
A better approximation is given by the following rule:
xi(ρ) = (1−√ρ)×xi(0) +√ρ×xi(1)
Figure 2: Simulation of the weightx1 when the correlation is constant
systematic (or market) risk, which is represented here by the portfoliox. It means that the risk contributionRCi is equal toxiβiσ(x). It follows that:
bjxiβi=bixjβj
We finally deduce that:
xi= biβi−1 Pn
j=1bjβj−1 (7)
The weight allocated to the componentiis thus inversely proportional to its beta. However, contrary to the solutions (4), (5) and (6), this one is endogenous sinceβi is the beta of the assetiwith respect to the RB portfolio11.
2.3.3 Existence and uniqueness of the RB portfolio
As for the ERC portfolio, we may use the SQP (Sequential Quadratic Programming) algo- rithm to find the RB portfolio12:
x⋆ = arg min
n
X
i=1
xi(Σx)i Pn
j=1xj(Σx)j −bi
!2
(8) u.c. 1⊤x= 1and0≤x≤1
11With the equation (7), we have a new interpretation of the functionfi(ρ, b):
fi(ρ, b) = bi
βi
σi
It is the product of the risk budget and the volatility scaled by the beta.
12From a numerical point of view, it is preferable to solve the system without the constraint1⊤x= 1and then to rescale the solution.
An alternative to the previous algorithm is to consider the following optimization problem:
y⋆ = arg minp
y⊤Σy (9)
u.c.
Pn
i=1bilnyi≥c y≥0
withcan arbitrary constant. In this case, the program is similar to a variance minimization problem subject to a constraint. The constantc could be adjusted to obtain Pn
i=1yi∗ = 1 implying that the RB portfolio is expressed asx⋆i =y∗i/Pn
i=1y∗i (see Appendix A.2).
The formulation (9) is very interesting because it demonstrates that the RB portfolio specified by the mathematical system (3) exists and is unique as long as the covariance matrix Σis positive-definite. Indeed, it corresponds to the minimization program of a quadratic function (a convex function) with a convex constraint13.
2.3.4 Comparing the risk budgeting portfolio with the weight budgeting port- folio
Maillardet al. (2010) shows that the ERC portfolio is located between the minimum vari- ance portfolio and the equally-weighted portfolio. We could extend this result to the risk budgeting portfolio. Letxwb be the weight budgeting (WB) porfolio, that is xi =bi. We have:
xi/bi=xj/bj (wb)
∂xiσ(x) =∂xjσ(x) (mv) xi∂xiσ(x)/bi=xj∂xjσ(x)/bj (rb)
Thus, a RB portfolio may be viewed as a portfolio located between the MV and WB port- folios. To elaborate further this point of view, let us consider a modified version of the optimization problem (9):
x⋆(c) = arg min√
x⊤Σx (10)
u.c.
Pn
i=1bilnxi≥c 1⊤x= 1
x≥0
In order to get the RB portfolio, one minimizes the volatility of the portfolio subject to an additional constraint,Pn
i=1bilnxi ≥c wherec is a constant being determined by the RB portfolio. In Appendix A.3, we prove that the volatilityσ(x⋆(c))is an increasing function ofc. Two polar cases can be defined withc=−∞for which one gets the MV portfolio and c=Pn
i=1bilnbi where one gets the WB portfolio. We finally deduce that the volatility of the risk budgeting portfolio is between the volatility of the minimum variance portfolio and the volatility of the weight budgeting portfolio:
σmv≤σrb≤σwb
2.3.5 Optimality
Like the ERC portfolio, the risk budgeting approach is an heuristic asset allocation method.
In finance, we like to derive optimal portfolios from the maximisation problem of an utility function. For example, Maillardet al. (2010) show that the ERC portfolio corresponds to
13Becausebi>0.
the tangency portfolio when the correlation is the same and when the assets have the same Sharpe ratio. For the RB portfolio, it is more difficult to find such properties14.
Let us consider the quadratic utility functionx⊤µ−φx⊤Σxcorresponding to the Markowitz criteria. The portfolioxis optimal if the vector of expected returns satisfies this relation- ship15:
˜ µ= 2
φΣx
If the RB portfolio is optimal, the performance contribution PCi of the asset i is then proportional to its risk budgets:
PCi = xiµ˜i
= 2
φxi(Σx)i
∝ bi
The specification of the risk budgets allows us to decide not only which amount of risk to invest in one asset, but also which amount of expected performance to attribute to this asset.
We consider a universe of 4 risky assets. Volatilities are respectively10%,20%,30%and 40%and the correlation matrix is equal to:
ρ=
1.00 0.80 1.00 0.20 0.20 1.00 0.20 0.20 0.50 1.00
We suppose that the risk budgets are inversely proportional to the volatilities andφ= 0.4.
We verify that performance contributions are proportional to risk budgets:
i xi ∂xiσ(x) RCi µ˜i PCi
1 64.9% 8.8% 48.0% 5.2% 48.0%
2 17.2% 16.6% 24.0% 9.8% 24.0%
3 11.2% 16.9% 16.0% 10.0% 16.0%
4 6.7% 21.4% 12.0% 12.7% 12.0%
2.3.6 Back to the case when some risk budgets are set to zero
The analysis of the optimization problem (10) permits also to clarify the disturbing point discussed in Section 2.2 about the possibility of several solutions when some assets have a null risk budget. By construction, the formulation (10) implies necessarily that the solution exists and is unique even ifbi = 0 for some assets. In Appendix A.2, we have specify this solution. LetN be the set of assets such that bi = 0. The solutionS1 of the optimization problem (10) satisfies the following relationships:
RCi=xi·∂xiσ(x) =bi if i /∈ N
xi = 0and∂xiσ(x)>0 (i) or
xi>0and ∂xiσ(x) = 0 (ii)
if i∈ N
14The RB portfolio could be written as the solution of a maximisation problem with an objective function equal toln“
Qn i=1xbii”
. Interpreting this problem as a utility maximization problem is attractive, but the objective function does not present right properties to be a utility function.
15We use the notation µ˜ to specify that it is the market price of expected returns with respect to the current portfolio.
The conditions (i) and (ii) are mutually exclusive for one asseti∈ N, but not necessarily for all the assetsi∈ N.
If we consider the system (2) such thatbi = 0for i∈ N, we can clarify the number of solutions by just analysing the solution of the optimization problem (10). LetN =N1F
N2 whereN1is the set of assets verifying the condition (i) andN2 is the set of assets verifying the condition (ii). The number of solutions16is equal to2mwherem=|N1|is the cardinality ofN1. Indeed, it is the number ofk-combinations for allk= 0,1, . . . , m.
Suppose thatm= 0. In this case, the solutionS1verifiesxi= 0ifbi= 0. It corresponds to the solution expected by the investor. Ifm≥1, there are several solutions, in particular the solutionS1given by the optimization problem (10) and another solutionS2withxi = 0 for all assets such thatbi= 0. We then obtain a paradox because even if S2is the solution expected by the investor, the only acceptable solution isS1. The argument is very simple.
Suppose that you imposebi =εi withεi >0 a small number for i∈ N. In this case, you obtain a unique solution. Ifεi →0, this solution will converge toS1, not to S2 or all the others solutionsSj forj= 3, . . . ,2m.
Table 2: Solution when the risk budgetb3 is equal to 0
ρ1,3=ρ2,3 Solution 1 2 3 σ(x)
−25%
xi 20.00% 40.00% 40.00%
S1 ∂xiσ(x) 16.58% 8.29% 0.00% 6.63%
RCi 50.00% 50.00% 0.00%
xi 33.33% 66.67% 0.00%
S2 ∂xiσ(x) 17.32% 8.66% −1.44% 11.55%
RCi 50.00% 50.00% 0.00%
xi 19.23% 38.46% 42.31%
S1′ ∂xiσ(x) 16.42% 8.21% 0.15% 6.38%
RCi 49.50% 49.50% 1.00%
25%
xi 33.33% 66.67% 0.00%
S1 ∂xiσ(x) 17.32% 8.66% 1.44% 11.55%
RCi 50.00% 50.00% 0.00%
Let us illustrate this paradox with the example given in Section 2.2. Results are reported in Table 2 when the risk budgets areb1 = 50%, b2 = 50% and b3 = 0%. If ρ1,3 =ρ2,3 =
−25%, we obtain two solutions S1 and S2. For the first solution, we have x1 = 20%, x2 = 40% and x3 = 40%, whereas the weights for the second solution are respectively x1 = 33.33%, x2 = 66.67% and x3 = 0%. We verify that the marginal risk ∂x3σ(x) is negative for the second solution indicating that it could not be a solution of the optimization problem. That’s why the volatility of this second solution (11.55%) is larger than the volatility to the first solution (6.63%). We also notice that if we perturb slightly the risk budgets (b1= 49.5%, b2 = 49.5%andb3= 1%), the solutionS1′ is closer to S1 than to S2. It confirms that the first solutionS1 is more acceptable. If all the correlations are positive (ρ1,3 = ρ2,3 = 25%), we do not face this problem. The unique solution is x1 = 33.33%, x2 = 66.67% and x3 = 0% and we verify that the marginal risk ∂x3σ(x) is positive. In
16To be more correct,2mis the maximum number of solutions. But the case when fixing some weights of N1to zero implies that other assets ofN1have a weight equal to zero has a small probability to occur.
Figure 3, we have represented the evolution of the volatility of the portfolio(50%,50%, x3) with respect to the weight x3. If ρ1,3 = ρ2,3 = −25%, the volatility is decreasing around x3 = 0meaning that the minimum is after x3 = 0. If ρ1,3 = ρ2,3 = 25%, the volatility is increasing aroundx3= 0meaning that the minimum is beforex3= 0. This figure illustrates that the solution verifiesx3>0 in the first case andx3= 0in the second case.
Figure 3: Evolution of the volatility with respect tox3
Let us add a fourth asset to the previous universe with σ4 = 10%,ρ1,4 =ρ2,4=−25%
andρ3,4 = 50%. Table 3 gives the results when the risk budgets are b1 = 50%, b2 = 50%, b3= 0%andb4= 0%. If we analyze the solutionS1, we notice that the numbermof assets such that both the marginal risk and the risk budgets are zero is equal to 2. That’s why we obtain22 = 4solutions. Another way to determine the number of solutions is to compute the solutionS2 desired by the investor and set mequal to the number of assets such that the marginal risk is strictly negative when the risk budget is equal to zero.
2.4 Generalization to other risk measures
We could show that some previous results are valid to convex risk measuresRthat satisfy the Euler decomposition, because it suffices to replace the marginal volatility by the marginal risk in the mathematical proofs. The optimization problem (9) becomes:
y⋆ = arg minR(y) u.c.
Pn
i=1bilnyi≥c y≥0
It is then easy to show that if the risk budgetsbiare strictly positive, the RB portfolio exists and is unique whereas there may be several solutions if some risk budgets are null. Using
Table 3: Solution when the risk budgetsb3andb4are equal to 0
Solution 1 2 3 4 σ(x)
xi 20.00% 40.00% 26.67% 13.33%
S1 ∂xiσ(x) 16.33% 8.16% 0.00% 0.00% 6.53%
RCi 50.00% 50.00% 0.00% 0.00%
xi 33.33% 66.67% 0.00% 0.00%
S2 ∂xiσ(x) 17.32% 8.66% −1.44% −2.89% 11.55%
RCi 50.00% 50.00% 0.00% 0.00%
xi 20.00% 40.00% 40.00% 0.00%
S3 ∂xiσ(x) 16.58% 8.29% 0.00% −1.51% 6.63%
RCi 50.00% 50.00% 0.00% 0.00%
xi 25.00% 50.00% 0.00% 25.00%
S4 ∂xiσ(x) 16.58% 8.29% −0.75% 0.00% 8.29%
RCi 50.00% 50.00% 0.00% 0.00%
the same proof by replacing the marginal volatility by the marginal risk, we could also show that the risk of the RB portfolio is between the risk of the minimum risk portfolio and the risk of the weight budgeting portfolio:
R(xmin)≤ R(xrb)≤ R(xwb)
And if the RB portfolio is optimal in the sense of the following utility function U(x) = x⊤µ−φR2(x), the performance contribution is equal to the risk contribution.
3 Some illustrations
The risk budgeting approach is an allocation method extensively used in asset management.
There is also a huge literature on this subject (see for example the books of Rahl (2000), Grinold and Kahn (2000), Meucci (2005) or Scherer (2007)). For a very long time, the risk budgeting technique has been applied to a universe of multi-assets classes to manage and monitor the portfolio risk of large and sophisticated institutional investors like pension funds (Sharpe, 2002). More recently, this technique has been used to build alternative indexes in order to provide new benchmarks than the traditional market-cap indexes.
3.1 Comparing risk budgeting portfolios with mean-variance or other heuristic portfolios
In this section, we first compare RB portfolios with mean-variance optimized (MVO) port- folios. Then, we give new interpretation of some heuristic portfolios as RB portfolios.
3.1.1 What are the main differences between RB and MVO portfolios?
The main difference between RB and MVO portfolios is that the last ones are based on optimization techniques. It implies that MVO portfolios are very sensitive to the inputs:
“The indifference of many investment practitioners to mean-variance optimiza- tion technology, despite its theoretical appeal, is understandable in many cases.
The major problem with MV optimization is its tendency to maximize the effects of errors in the input assumptions. Unconstrained MV optimization can yield results that are inferior to those of simple equal-weighting schemes” (Michaud, 1989).
For MVO portfolios, the risk approach ismarginal and the quantity of interest to study is the marginal volatility. For RB portfolios, the risk approach becomesglobal by mixing the marginal volatility and the weight. Let us illustrate these two ways to consider risk by an example. We consider three assets withµ1= 8%,µ2= 8%,µ3= 5%,σ1= 20%,σ2= 21%, σ3 = 10%. We assume that the correlation between the asset returns is uniform and is equal to80%. The MVO portfolio is equal tox1= 38.3%,x2= 20.2%andx3= 41.5%if we target a portfolio volatility of15%. In this case, the risk budgets are respectivelyb1= 49.0%, b2= 25.8%andb3= 25.2%. Of course, the RB portfolio corresponding to these risk budgets is exactly the MVO portfolio. In Table 4, we have reported how these two portfolios evolve when we change slightly the inputs17. For example, if the uniform correlation is90%, the MVO and RB portfolios become(44.6%,8.9%,46.5%)and(38.9%,20.0%,41.1%). The MVO portfolio is thus very sensitive to the input parameters whereas the RB portfolio is more robust. In a dynamic investment strategy, the input parameters will change from one period to another period. Because (µt+1,Σt+1)will certainly be different from (µt,Σt), it would imply that a dynamic strategy based on MVO portfolios will generate a higher turnover than a dynamic strategy based on RB portfolios. This lack of robustness penalizes so much MVO portfolios that they are not used in practice without introducing some constraints.
Table 4: Sensitivity of MVO and RB portfolios to input parameters
ρ 70% 90% 90%
σ2 18% 18%
µ1 9%
x1 38.3% 38.3% 44.6% 13.7% 0.0% 56.4%
MVO x2 20.2% 25.9% 8.9% 56.1% 65.8% 0.0%
x3 41.5% 35.8% 46.5% 30.2% 34.2% 43.6%
x1 38.3% 37.7% 38.9% 37.1% 37.7% 38.3%
RB x2 20.2% 20.4% 20.0% 22.8% 22.6% 20.2%
x3 41.5% 41.9% 41.1% 40.1% 39.7% 41.5%
The problem of MVO portfolios comes from the solution structure, because optimal solutions are of the following form:
x⋆∝Σ−1µ
The important quantity is thenI = Σ−1, which is called the information matrix. We know that the eigendecomposition ofI is related to the eigendecomposition of Σin the following way: the eigenvectors are the same and the eigenvalues ofI are equal to the inverse of the eigenvalues ofΣ:
Vi(I) = Vn−i(Σ) λi(I) = 1
λn−i(Σ)
For example, if we consider our previous example, we obtain the results in Table 5. It implies
17For the MVO portfolio, the objective is to maximize the expected return for a given volatility of 15%, whereas the objective of the RB portfolio is to match the risk budgets(49.0%,25.8%,25.2%).
Table 5: Eigendecomposition of the covariance and information matrices Covariance matrixΣ Information matrixI
Asset / Factor 1 2 3 1 2 3
1 65.35% −72.29% −22.43% −22.43% −72.29% 65.35%
2 69.38% 69.06% −20.43% −20.43% 69.06% 69.38%
3 30.26% −2.21% 95.29% 95.29% −2.21% 30.26%
Eigenvalue 8.31% 0.84% 0.26% 379.97 119.18 12.04
% cumulated 88.29% 97.20% 100.00% 74.33% 97.65% 100.00%
that MVO portfoliosx⋆ depends also on the most important factors of I, that is the less important factors ofΣ. But the last eigenfactors of a covariance matrix represent generally noise or very specific factors. It explains why MVO portfolios are not robust to input parameters, because a small change in the covariance matrix will change dramatically the last factor. To solve this type of problem, market practitioners introduce some regularization techniques:
• regularization of the objective function by using resampling techniques (Tütüncü and Koenig, 2004);
• regularization of the covariance matrix:
– Factor analysis based on PCA, ICA, etc. (Hyvärinen and Oja, 2000);
– Shrinkage methods (Ledoit and Wolf, 2003);
– Random matrix theory (Laloux et al., 1999);
– etc.
• regularization of the program specification by introducing some constraints on weights.
Even if the first two techniques are used, practitioners generally add constraints on weights in order to obtain a more satisfactory solution from a financial point of view. Jagannathan and Ma (2003) have shown that it is equivalent to shrink the covariance matrix. Introducing some constraints on weights may also lead to a covariance matrix that could be very different from the original one (Roncalli, 2011). With RB portfolios, we don’t need to add constraints to obtain a satisfactory solution. It is certainly the main difference between MVO and RB portfolios.
3.1.2 New interpretation of the EW, MV, MDP and ERC portfolios
The risk budgeting approach leads us to a new interpretation of some specific portfolios:
• The equally-weighted (EW) portfolio could be viewed as a risk budgeting portfolio when the risk budget is proportional to the beta of the asset. Using the result (7), bi=βi/nimplies that:
xi= biβi−1 Pn
j=1bjβj−1 = 1 n
• The minimum variance portfolio is a risk budgeting portfolio when the risk budget is equal to the weight of the asset:
bi=xi
Let us consider an iterated portfolio
x(t)1 , . . . , x(t)1
wheret represents the iteration.
The portfolio is defined such that the risk budget b(t)i of the asset i at iteration t corresponds to the weight x(t−1)i at iteration t−1. If the portfolio
x(t)1 , . . . , x(t)1 admits a limit whent→ ∞, it is equal to the minimum variance portfolio.
• The MDP portfolio of Choueifaty and Coignard (2008)et al. (2010) is the RB portfolio such that the risk budgets are proportional to the product of the weight of the asset and its volatility18:
bi∝xiσi
• Maillard et al. (2010) defines the ERC portfolio as the portfolio such that the risk contribution of all assets is the same. Using the previous results, it is also the optimal portfolio such that the performance contribution of all assets is the same.
Let us consider an example with 5 assets. The volatilities are respectively σ1 = 10%, σ2= 20%,σ3= 30%,σ4= 40%, and σ5= 30%whereas the correlation matrix is equal to:
ρ=
1.00 0.80 1.00
0.00 0.00 1.00 0.00 0.00 −0.50 1.00 0.00 0.00 −0.20 0.80 1.00
Results are reported in Table 6. The second columnxiindicates the weight of assetiin the portfolio,∂xiσ(x)corresponds to its marginal volatility,RCi is its risk contribution andβi
is its beta. VCi =xiσi corresponds to the volatility contribution. It could be interpreted as the risk contribution of the asset in the case of a perfect correlation (ρi,j = 1) between asset returns:
VCi = xi(Σx)i
√x⊤Σx
= xiσi
Pn j=1xjσj
r
Pn j=1xjσj
2
= xiσi
The seventh column gives the market price (or implied expected return)µ˜i of the asset19 whereas the last column indicates the performance contributionPCi=xiµ˜i/
Pn j=1xjµ˜j
if the portfolio is optimal.
We verify that the risk contributions of the EW portfolio are proportional to the beta, that those of the MV20 and MDP portfolios are equal respectively to the weights and the volatility contributions and that the performance contributions of the ERC portfolio are equal. This simple example shows how different could be these heuristic portfolios in terms of weights but also in terms of risk metrics (risk contribution, beta, etc.).
18As for the MV portfolio, this solution is endogenous because it depends on the weights of the portfolio.
19It has been calibrated in order that the average market price of the five assets is equal to8%.
20If we consider the iterated procedure presented above to find the MV portfolio with the EW portfolio as the starting portfolio, the convergence is achieved after 13 iterations for a tolerance of 1 bp in terms of volatility or mean square error.
Table 6: EW, MV, MDP and ERC portfolios
i xi ∂xiσ(x) RCi βi VCi µ˜i PCi
1 20.00% 3.84% 5.68% 0.28 7.69% 2.27% 5.68%
2 20.00% 8.27% 12.23% 0.61 15.38% 4.89% 12.23%
EW 3 20.00% 1.77% 2.62% 0.13 23.08% 1.05% 2.62%
4 20.00% 28.96% 42.79% 2.14 30.77% 17.12% 42.79%
5 20.00% 24.82% 36.68% 1.83 23.08% 14.67% 36.68%
1 74.46% 8.63% 74.46% 1.00 46.31% 7.14% 74.46%
2 0.00% 13.81% 0.00% 1.60 0.00% 11.43% 0.00%
MV 3 14.93% 8.63% 14.93% 1.00 27.85% 7.14% 14.93%
4 9.71% 8.63% 9.71% 1.00 24.16% 7.14% 9.71%
5 0.90% 8.63% 0.90% 1.00 1.68% 7.14% 0.90%
1 27.78% 4.42% 10.87% 0.39 10.87% 2.94% 10.87%
2 13.89% 8.85% 10.87% 0.78 10.87% 5.88% 10.87%
MDP 3 33.33% 13.27% 39.13% 1.17 39.13% 8.82% 39.13%
4 25.00% 17.69% 39.13% 1.57 39.13% 11.76% 39.13%
5 0.00% 15.92% 0.00% 1.41 0.00% 10.59% 0.00%
1 35.55% 6.00% 20.00% 0.56 15.90% 3.85% 20.00%
2 17.77% 12.00% 20.00% 1.13 15.90% 7.69% 20.00%
ERC 3 22.14% 9.63% 20.00% 0.90 29.71% 6.17% 20.00%
4 12.47% 17.11% 20.00% 1.60 22.30% 10.97% 20.00%
5 12.07% 17.67% 20.00% 1.66 16.20% 11.32% 20.00%
3.2 Risk-based allocation
In the first part of this section, we present the concept of risk parity funds. In the second part, we show how risk budgeting allocation could be used to build the strategic asset allocation of long-term investors.
3.2.1 Risk parity funds
The business of diversified funds has suffered a lot of criticisms, both from a theoretical and a practical point of view. If we consider the modern theory of portfolio management, mean-variance portfolios of risky assets define the set of efficient portfolios, that is the efficient frontier of Markowitz (see Figure 4). If we introduce a risk-free asset, the efficient frontier becomes a straight line called thesecurity (or capital) market line. This frontier is a graphical representation of the risk-return profile of portfolios invested in the risk-free asset and a particular portfolio, also called the tangency portfolio. This last one is the portfolio which belongs to the efficient frontier and which maximises the Sharpe ratio. A consequence is that the portfolios of the security market line dominates the portfolios of the efficient frontier. The allocation choice between the cash and the tangency portfolio depends on the investor profile. Generally, we distinguish three profiles depending on the risk tolerance:
• Conservative (low risk tolerance)
• Moderate (medium risk tolerance)
• Agressive (high risk tolerance)
Figure 4: The asset allocation puzzle
In Figure 4, these three profiles have been represented in the top-right quadrant. Because the conservative investor has a smaller appetite for risk than the aggressive investor, its portfolio will contain more cash and less risky assets. But the relative proportions between risky assets are the same for the conservative and agressive investors. In practice, it is inefficient to pay fees just to leverage or deleverage the tangency portfolio. However, the business of diversified funds is largely based on this framework. Generally, we distinguish three profiles depending on the benchmark of the diversified fund:
• Defensive (80% of bonds and20% of equities)
• Balanced (50% of bonds and50% of equities)
• Dynamic (20%of bonds and80%of equities)
In this case, the distinction between the three fund profiles depends on the weight of equities of the corresponding benchmark. In Figure 4, these three profiles have been represented in the bottom quadrant. Contrary to the optimal portfolios corresponding to the three investor profiles, the portfolios corresponding to these three lifestyle funds are located on the efficient frontier and not on the security market line. The relationship between investor profiles and fund profiles is then not easy. The business of diversified funds suggests however that a defensive (resp. balanced or dynamic) fund profile matches the need of a conservative (resp.
moderate or agressive) investor profile. But we clearly face a gap between the theory and the industry practice. This paradox is called in finance theasset allocation puzzle21. From the practical point of view, the main criticisms come from the fact that the allocation of diversified funds is very static and does not justify the high level of management fees. More
21Even if we could partially solve this problem (Bajeux-Besnainouet al., 2003), the controversy is still relevant today (Campbell and Viciera, 2002).
recently, the industry has launched similar products called flexible funds in order to answer these criticisms.
In what follows, the bond and equity asset classes are represented by theCitigroup WGBI All Maturities Index and theMSCI World TR Net Index. In Figure 5, we have simulated the evolution of the risk contribution of these two asset classes for the three diversified funds with a one-year rolling empirical covariance matrix. We notice that these risk contributions are time-varying, especially for defensive and balanced funds. For defensive funds, the bond asset class has a larger risk contribution than the equity asset class. For balanced funds, we obtain the opposite, meaning that even if they are very well balanced in terms of weights, they are certainly not in terms of risk diversification. For dynamic funds, the risk is almost entirely explained by equities. In some sense, dynamic funds may be viewed as a deleverage of an equity exposure. We also notice that there is no mapping between fund profiles and volatility regimes. For example, the volatility of a dynamic fund in 2006 is smaller than the volatility of a balanced fund in 2009.
Figure 5: Risk contributions of diversified funds
These drawbacks lead the investment industry to propose an alternative to these diver- sified funds. A risk parity (RP) fund is an ERC strategy on multi-assets classes:
“Diversify, but diversify by risk, not by dollars–that is, take a similar amount of risk in equities and in bonds” (Asness et al., 2012).
Applying this concept to our example is equivalent to build a portfolio where the risk of the bond asset class is equal to the risk of the equity asset class. If we assume that we rebalance the portfolio every month, we obtain the dynamic allocation given in Figure 6.
The simulated performance of this risk parity fund is reported in the bottom-right quadrant.
It is difficult to compare it to those of the diversified funds (bottom-left quadrant), but we
Figure 6: Comparison of diversified and risk parity funds
notice that risk parity and defensive funds are close. We have also reported the simulated performance when the weights are constant and are equal to the average weights of the risk parity fund. Compared to this static fund, the RP fund presents an outperformance of 40 bps and a volatility smaller than 35 bps. Sometimes, the ERC strategy is combined with a leverage effect in order to obtain a more risky profile. For example, if we apply a10%
volatility target, we obtain the performance of the leveraged risk parity fund (bottom-right quadrant). The leverage RP fund has a volatility similar to the balanced fund, but a better performance.
Remark 3 In practice, risk parity funds use a larger universe than the example presented here. It could be exposures on American, European, Japanese and Emerging Markets equities, large cap and small cap equities, American and European sovereign bonds, inflation-linked, corporate and high yield bonds, etc.
3.2.2 Strategic asset allocation
Strategic asset allocation (SAA) is the choice of equities, bonds, and alternative assets that the investor wishes to hold for the long-run, usually from 10 to 50 years. Combined with tactical asset allocation (TAA) and constraints on liabilities, it defines the long-term in- vestment policy of pension funds. By construction, SAA requires long-term assumptions of asset risk/return characteristics as a key input. It could be done using macroeconomic models and forecasts of structural factors such as population growth, productivity and in- flation (Eychenneet al., 2011). Using these inputs, one may obtain a SAA portfolio using a mean-variance optimization procedure. Because of the uncertainty of these inputs and the instability of mean-variance portfolios, a lot of institutional investors prefer to use these fig- ures as a criteria to select the asset classes they would like to have in their strategic portfolio and to define the corresponding risk budgets.
Let us illustrate this process with an example. We consider a universe of nine asset classes : US Bonds 10Y (1), EURO Bonds 10Y (2), Investment Grade Bonds (3), High Yield Bonds (4), US Equities (5), Euro Equities (6), Japan Equities (7), EM Equities (8) and Commodities (9). In Table 7, we indicate the long-run statistics used to compute the strategic asset allocation22. Based on these statistics and its constraints, the pension fund decides to define its strategic portfolio according to the risk budgets given in Figure 7.
Table 7: Expected returns, risks and correlations (in %)
µi σi ρi,j
(1) (2) (3) (4) (5) (6) (7) (8) (9) (1) 4.2 5.0 100
(2) 3.8 5.0 80 100
(3) 5.3 7.0 60 40 100
(4) 10.4 10.0 −20 −20 50 100 (5) 9.2 15.0 −10 −20 30 60 100
(6) 8.6 15.0 −20 −10 20 60 90 100
(7) 5.3 15.0 −20 −20 20 50 70 60 100
(8) 11.0 18.0 −20 −20 30 60 70 70 70 100
(9) 8.8 30.0 0 0 10 20 20 20 30 30 100
Figure 7: Risk budgeting policy of the pension fund
✬
✫
✩
✪ 100%
45%
Bonds
45% Equities
10%
Alternative Commodities 10%
EM Equities Japan Equities EURO Equities US Equities
15%
0%
10%
20%
HY Bonds IG Bonds EURO Bonds US Bonds
0%
15%
10%
20%
If we match these risk budgets, we obtain the solution RB given in Table 8. Of course, the pension fund may modify this strategic portfolio by using the expected returns. It could be done in Black-Litterman or tracking error frameworks. For example, if we would like to maximize the expected return of the portfolio according to a1%tracking error with respect to the RB portfolio, we obtain the RB⋆ portfolio given in Table 8. We could compare this modified portfolio with the mean-variance optimized (MVO) portfolio which has the same ex-ante volatility. Results are reported in Table 8 and in Figure 8. First, we notice that the two portfolios RB⋆ and MVO are very close in terms of risk/return profile. Second, the RB⋆ portfolio is much more diversified than the MVO portfolio, which concentrates50% of its risk in the EM Equities asset class. The MVO portfolio is too far from the pension fund objective in terms of risk budgeting to be an acceptable strategic portfolio.
22This figures are taken from Eychenneet al. (2011) and Eychenne and Roncalli (2011)
Figure 8: Efficient frontier
Table 8: Long-term strategic portfolios
Asset class RB RB⋆ MVO
xi RCi xi RCi xi RCi
(1) 36.8% 20.0% 45.9% 18.1% 66.7% 25.5%
(2) 21.8% 10.0% 8.3% 2.4% 0.0% 0.0%
(3) 14.7% 15.0% 13.5% 11.8% 0.0% 0.0%
(5) 10.2% 20.0% 10.8% 21.4% 7.8% 15.1%
(6) 5.5% 10.0% 6.2% 11.1% 4.4% 7.6%
(8) 7.0% 15.0% 11.0% 24.9% 19.7% 49.2%
(9) 3.9% 10.0% 4.3% 10.3% 1.5% 2.7%
3.3 Risk-based indexation
More recently, risk-budgeting techniques have been considered to build alternative bench- marks to market-cap indexes. These last ones have been particularly criticized by academics and market professionals and we observe a growing interest by sophisticated institutional investors.
3.3.1 Equity indexation
Capitalization-weighted indexation is the most common way to gain access to broad equity market performance. It is often backed by results of modern portfolio theory because it is as- similated to the market portfolio. Moreover, it provides two main advantages: simplicity of management (low turnover and transaction costs) and ease of understanding and replication.
However, it also presents some drawbacks. For example, capitalization-weighted indexation is by definition a trend-following strategy where momentum bias leads to bubble risk expo- sure as weights of best performers increase. Moreover, the absence of portfolio construction rules leads to concentration issues (in terms of sectors or stocks). In this context, the con- cept of alternative-weighted indexation emerged after the dot.com bubble. An alternative- weighted index is defined as an index in which assets are weighted differently than in the market capitalization approach. We generally distinguish two forms of alternative-weighted indexation: fundamental and risk-based. Fundamental indexation defines the weights as a function of economic metrics like dividends or earnings, whereas risk-based indexation defines the weights as a function of individual and common risks.
Figure 9: Performance of Eurostoxx 50 and ERC Eurozone indexes since January 1993
By construction, the ERC method belongs to the second form of alternative-weighted indexation. In Figure 9, we have reported the performance of the Eurostoxx 50 NR index and the ERC Eurozone index23 from January 1993 to December 2011. The ERC Eurozone index has the same universe than the Eurostoxx 50 NR index but it is rebalanced every month following the ERC method. We notice that the ERC Eurozone index has a smaller volatility (21.2%versus22.9%) and a smaller drawdown (55.1%versus66.6%) than the Eu- rostoxx 50 index. What is more surprising is that the ERC Eurozone index outperforms the Eurostoxx 50 index (the yearly returns are respectively equal to10.7%and7.1%). One way
23The corresponding Bloomberg tickers are respectively SX5T and SGIXERC.
Table 9: Composition of Eurostoxx 50 and ERC Eurozone indexes (01/01/2012)
Name Eurostoxx 50 ERC Eurozone
xi RCi µ˜i PCi xi RCi µ˜i PCi PC⋆i
TOTAL SA 6.55 5.00 5.74 5.00 2.45 2.00 5.72 2.00 2.02 SANOFI 5.22 3.82 5.51 3.82 2.55 2.00 5.52 2.00 2.01 SIEMENS AG 4.77 4.24 6.68 4.24 2.11 2.00 6.65 2.00 2.02 TELEFONICA SA 3.98 3.46 6.55 3.46 2.16 2.00 6.52 2.00 2.02 BANCO SANTANDER SA 3.76 4.37 8.76 4.37 1.64 2.00 8.58 2.00 2.06 BASF SE 3.70 3.91 7.96 3.91 1.78 2.00 7.90 2.00 2.03 ENI SPA 3.17 2.82 6.71 2.82 2.09 2.00 6.69 2.00 2.01 UNILEVER NV 3.08 1.00 2.43 1.00 5.55 2.00 2.53 2.00 1.94 BAYER AG 3.05 2.91 7.17 2.91 1.97 2.00 7.12 2.00 2.02 SAP AG 2.82 1.81 4.82 1.81 2.89 2.00 4.86 2.00 1.99 ANHEUSER-BUSCH INB 2.63 1.21 3.46 1.21 3.95 2.00 3.55 2.00 1.96 ALLIANZ SE 2.50 3.46 10.41 3.46 1.36 2.00 10.32 2.00 2.03 E.ON AG 2.49 2.90 8.76 2.90 1.61 2.00 8.71 2.00 2.02 BBVA 2.41 3.14 9.82 3.14 1.46 2.00 9.64 2.00 2.05 BNP PARIBAS 2.26 4.00 13.30 4.00 1.08 2.00 13.04 2.00 2.05 DAIMLER AG 2.26 2.56 8.53 2.56 1.64 2.00 8.57 2.00 2.00 DANONE 2.22 1.15 3.89 1.15 3.53 2.00 3.98 2.00 1.97 LVMH 2.18 1.97 6.79 1.97 2.04 2.00 6.89 2.00 1.99 GDF SUEZ 2.10 2.22 7.93 2.22 1.76 2.00 7.94 2.00 2.01 DEUTSCHE BANK AG 2.05 3.19 11.71 3.19 1.21 2.00 11.56 2.00 2.04 AIR LIQUIDE SA 2.03 1.43 5.29 1.43 2.63 2.00 5.34 2.00 1.99 DEUTSCHE TELEKOM AG 1.97 1.41 5.38 1.41 2.58 2.00 5.44 2.00 1.99 FRANCE TELECOM SA 1.73 1.32 5.73 1.32 2.44 2.00 5.75 2.00 2.00 SCHNEIDER ELECTRIC SA 1.67 2.27 10.24 2.27 1.37 2.00 10.26 2.00 2.01 ING GROEP NV 1.60 3.00 14.08 3.00 1.00 2.00 14.03 2.00 2.02 VIVENDI 1.56 1.42 6.84 1.42 2.04 2.00 6.89 2.00 2.00 ENEL SPA 1.50 1.46 7.32 1.46 1.92 2.00 7.33 2.00 2.01 REPSOL YPF SA 1.45 1.48 7.68 1.48 1.84 2.00 7.64 2.00 2.02 L’OREAL 1.44 0.82 4.30 0.82 3.20 2.00 4.38 2.00 1.97 AXA SA 1.39 2.43 13.13 2.43 1.08 2.00 13.03 2.00 2.03 INTESA SANPAOLO 1.35 2.48 13.82 2.48 1.03 2.00 13.64 2.00 2.04 VINCI SA 1.34 1.45 8.16 1.45 1.71 2.00 8.19 2.00 2.00 IBERDROLA SA 1.32 1.28 7.30 1.28 1.93 2.00 7.27 2.00 2.02 VOLKSWAGEN AG 1.24 1.45 8.79 1.45 1.57 2.00 8.91 2.00 1.98 BMW AG 1.24 1.33 8.05 1.33 1.72 2.00 8.17 2.00 1.99 ASSICURAZIONI GENERALI 1.16 1.18 7.68 1.18 1.84 2.00 7.63 2.00 2.02 PHILIPS ELEC(KON) 1.14 1.08 7.09 1.08 1.97 2.00 7.14 2.00 2.00 MUENCHENER RUECKVER 1.14 1.08 7.14 1.08 1.96 2.00 7.15 2.00 2.01 NOKIA OYJ 1.06 1.01 7.21 1.01 1.89 2.00 7.42 2.00 1.95 INDITEX 1.00 0.52 3.93 0.52 3.44 2.00 4.09 2.00 1.94 SOCIETE GENERALE 1.00 2.06 15.55 2.06 0.92 2.00 15.29 2.00 2.05 RWE AG 0.98 1.20 9.23 1.20 1.52 2.00 9.22 2.00 2.02 ARCELORMITTAL 0.98 1.43 11.02 1.43 1.27 2.00 11.03 2.00 2.01 CIE DE SAINT-GOBAIN 0.96 1.29 10.13 1.29 1.38 2.00 10.19 2.00 2.00 UNIBAIL-RODAMCO SE 0.95 0.75 5.96 0.75 2.32 2.00 6.03 2.00 1.99 CRH PLC 0.83 1.00 9.01 1.00 1.54 2.00 9.11 2.00 1.99 UNICREDIT SPA 0.79 1.35 12.80 1.35 1.11 2.00 12.62 2.00 2.04 CARREFOUR SA 0.75 0.84 8.43 0.84 1.65 2.00 8.50 2.00 1.99 TELECOM ITALIA SPA 0.64 0.57 6.64 0.57 2.10 2.00 6.70 2.00 2.00 DEUTSCHE BOERSE AG 0.56 0.46 5.17 0.46 2.22 2.00 5.30 2.00 1.65
